Advertisements
Advertisements
Question
Match the statements of column A and B.
| Column A | Column B |
| (a) The value of 1 + i2 + i4 + i6 + ... i20 is | (i) purely imaginary complex number |
| (b) The value of `i^(-1097)` is | (ii) purely real complex number |
| (c) Conjugate of 1 + i lies in | (iii) second quadrant |
| (d) `(1 + 2i)/(1 - i)` lies in | (iv) Fourth quadrant |
| (e) If a, b, c ∈ R and b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are non real (complex) and |
(v) may not occur in conjugate pairs |
| (f) If a, b, c ∈ R and b2 – 4ac > 0, and b2 – 4ac is a perfect square, then the roots of the equation ax2 + bx + c = 0 |
(vi) may occur in conjugate pairs |
Solution
| Column A | Answers |
| (a) The value of 1+ i2 + i4 + i6 + ... i20 is | (ii) purely real complex number |
| (b) The value of `i^(-1097)` is | (i) purely imaginary complex number |
| (c) Conjugate of 1 + i lies in | (iv) Fourth quadrant |
| (d) `(1 + 2i)/(1 - i)` lies in | (iii) second quadrant |
| (e) If a, b, c ∈ R and b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are non real (complex) and |
(vi) may occur in conjugate pairs |
| (f) If a, b, c ∈ R and b2 – 4ac > 0, and b2 – 4ac is a perfect square, then the roots of the equation ax2 + bx + c = 0 |
(v) may not occur in conjugate pairs |
Explanation:
(a) Because 1 + i2 + i4 + i6 + ... i20
= 1 – 1 + 1 – 1 + ... + 1 = 1 ......(Which is purely a real complex number.)
(b) Because `i^(-1097)` = `1/((i)^1097)`
= `1/(i^(4 xx 274 + 1)`
= `1/((i^4)^274i)`
= `1/i`
= `i/i^2`
= –i
Which is purely imaginary complex number.
(c) Conjugate of 1 + i is 1 – i which is represented by the point (1, –1) in the fourth quadrant.
(d) Because `(1 + 2i)/(1 - i) = (1 + 2i)/(1 - i) xx (1 + i)/(1 + i)`
= `(-1 + 3i)/2`
= `-1/2 + 3/2 i`
Which is represented by the point `(- 1/2, 3/2)` in the second quadrant.
(e) If b2 – 4ac < 0 = D < 0 i.e., square root of D is a imaginary number.
Therefore, roots are x = `(-b +- "Imaginary Number")/(2a)`
i.e., roots are in conjugate pairs.
(f) Consider the equation `x^2 - (5 + sqrt(2)) x + 5 sqrt(2)` = 0, Where a = 1, b = `-(5 + sqrt(2))`, c = `5 sqrt(2)`, Clearly a, b, c ∈ R.
Now D = b2 – 4ac = `{- (5 + sqrt(2))}^2 - 4.1.5 sqrt(2) = (5 - sqrt(2))^2`.
Therefore x = `(5 + sqrt(2) +- 5 - sqrt(2))/2` = `5sqrt(2)` which do not form a conjugate pair.
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: i9 + i19
Express the given complex number in the form a + ib: `(1/5 + i 2/5) - (4 + i 5/2)`
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Find the value of the following expression:
i5 + i10 + i15
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
Express the following complex number in the standard form a + i b:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
If z1 and z2 are two complex numbers such that \[\left| z_1 \right| = \left| z_2 \right|\] and arg(z1) + arg(z2) = \[\pi\] then show that \[z_1 = - \bar{{z_2}}\].
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].
Disclaimer: There is a misprinting in the question. It should be \[\left( 1 + i\sqrt{3} \right)\] instead of \[\left( 1 + \sqrt{3} \right)\].
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
The argument of \[\frac{1 - i}{1 + i}\] is
The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`(2 + sqrt(-3))/(4 + sqrt(-3))`
Evaluate the following : i93
Evaluate the following : `1/"i"^58`
Evaluate the following : i–888
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
If z1 and z2 both satisfy `z + barz = 2|z - 1|` arg`(z_1 - z_2) = pi/4`, then find `"Im" (z_1 + z_2)`.
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
Show that `(-1 + sqrt3 "i")^3` is a real number.
Show that `(-1+ sqrt(3)i)^3` is a real number.
